Non-uniform spacings processes

We provide a joint strong approximation of the uniform spacings empirical pro1 cess and of the uniform quantile process by sequences of independent Gaussian processes. 2 This allows us to obtain an explicit description of the limiting Gaussian process generated by 3 the sample spacings from a non-uniform distribution. It is of the form B(t)+ (1− σF ){(1− 4 t) log(1 − t)} ∫ 1 0 {B(s)/(1 − s)}ds, for 0 ≤ t ≤ 1, where {B(t) : 0 ≤ t ≤ 1} denotes a 5 Brownian bridge, and where σ 2 F = Var(log f (X)) is a factor depending upon the underlying 6 distribution function F(·) = P(X ≤ x) through its density f (x) = d dx F(x). We provide 7 a strong approximation of the non-uniform spacings processes by replicæ of this Gaussian 8 process, with limiting sup-norm rate OP(n−1/8(log n)1/2). The limiting process reduces to a 9 Brownian bridge if and only if σ 2 F = 1, which is the case when the sample observations are 10 exponential. For uniform spacings, we get σ 2 F = 0, which is in agreement with the results of 11 Beirlant (In: Limit theorems in probability and statistics, Proc Coll Math Soc J Bolyai, vol 12 36, Akadémiai Kiadó, Budapest, pp 77–80, 1984), and Aly et al. (Z Wahrsch Verw Gebiete 13 66:461–484, 1984). 14